Bisexual Galton-Watson branching process with population-size-dependent mating

2002 ◽  
Vol 39 (3) ◽  
pp. 479-490 ◽  
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Population Size ◽  
Generating Functions ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Size Dependent ◽  
Probability Generating Functions ◽  
Necessary And Sufficient ◽  
Mating Function

In this paper, we introduce a bisexual Galton-Watson branching process with mating function dependent on the population size in each generation. Necessary and sufficient conditions for the process to become extinct with probability 1 are investigated for two possible conditions on the sequence of mating functions. Some results for the probability generating functions associated with the process are also given.

Bisexual Galton-Watson branching process with population-size-dependent mating

2002 ◽  
Vol 39 (03) ◽  
pp. 479-490 ◽  
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Population Size ◽  
Generating Functions ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Size Dependent ◽  
Probability Generating Functions ◽  
Necessary And Sufficient ◽  
Mating Function

In this paper, we introduce a bisexual Galton-Watson branching process with mating function dependent on the population size in each generation. Necessary and sufficient conditions for the process to become extinct with probability 1 are investigated for two possible conditions on the sequence of mating functions. Some results for the probability generating functions associated with the process are also given.

scholarly journals Criterion for unlimited growth of critical multidimensional stochastic models

2016 ◽  
Vol 48 (4) ◽  
pp. 972-988 ◽  
Author(s):  
Etienne Adam
Keyword(s):  
Large Class ◽  
Stochastic Models ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Positive Probability ◽  
Size Dependent ◽  
Unlimited Growth ◽  
Necessary And Sufficient

AbstractWe give a criterion for unlimited growth with positive probability for a large class of multidimensional stochastic models. As a by-product, we recover the necessary and sufficient conditions for recurrence and transience for critical multitype Galton–Watson with immigration processes and also significantly improve some results on multitype size-dependent Galton–Watson processes.

On the L 2-convergence of a superadditive bisexual Galton-Watson branching process

1997 ◽  
Vol 34 (03) ◽  
pp. 575-582 ◽  
Author(s):  
M. González ◽  
M. Molina
Keyword(s):  
Branching Process ◽  
Final Section ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

In this paper the L 2-convergence of a superadditive bisexual Galton–Watson branching process is studied. Necessary and sufficient conditions for the convergence of the suitably normed process are given. In the final section, a result about one of the most important bisexual models is proved.

On some classes of population-size-dependent Galton–Watson processes

1985 ◽  
Vol 22 (01) ◽  
pp. 25-36
Author(s):  
Reinhard Höpfner
Keyword(s):  
Population Size ◽  
Lower Bounds ◽  
Generating Functions ◽  
Transition Probability ◽  
Limit Distributions ◽  
Size Dependent ◽  
Probability Generating Functions ◽  
Iteration Methods ◽  
Functional Iteration

Some classes of population-size-dependent Galton–Watson processes {Z(t)} t=0,1, …, whose transition probability generating functions allow for certain upper or lower bounds, can be treated by means of functional iteration methods. Criteria for almost certain extinction are obtained as well as gammatype limit distributions for Z(t)/t as t → ∞ the results can be stated under conditions on moments of the reproduction distributions.

On the class of controlled branching processes with random control functions

2002 ◽  
Vol 39 (4) ◽  
pp. 804-815 ◽  
Author(s):  
M. González ◽  
M. Molina ◽  
I. Del Puerto
Keyword(s):  
Population Size ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Control Functions ◽  
Controlled Branching Processes ◽  
Necessary And Sufficient ◽  
Random Control

In this paper, the class of controlled branching processes with random control functions introduced by Yanev (1976) is considered. For this class, necessary and sufficient conditions are established for the process to become extinct with probability 1 and the limit probabilistic behaviour of the population size, suitably normed, is investigated.

On some classes of population-size-dependent Galton–Watson processes

1985 ◽  
Vol 22 (1) ◽  
pp. 25-36 ◽  
Author(s):  
Reinhard Höpfner
Keyword(s):  
Population Size ◽  
Lower Bounds ◽  
Generating Functions ◽  
Transition Probability ◽  
Limit Distributions ◽  
Size Dependent ◽  
Probability Generating Functions ◽  
Iteration Methods ◽  
Functional Iteration

Some classes of population-size-dependent Galton–Watson processes {Z(t)}t=0,1, …, whose transition probability generating functions allow for certain upper or lower bounds, can be treated by means of functional iteration methods. Criteria for almost certain extinction are obtained as well as gammatype limit distributions for Z(t)/t as t → ∞ the results can be stated under conditions on moments of the reproduction distributions.

scholarly journals Extinction Probability of Interacting Branching Collision Processes

2012 ◽  
Vol 44 (1) ◽  
pp. 226-259 ◽  
Author(s):  
Anyue Chen ◽  
Junping Li ◽  
Yiqing Chen ◽  
Dingxuan Zhou
Keyword(s):  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Extinction Probability ◽  
Hitting Times ◽  
Collision Process ◽  
Markov Branching Process ◽  
Extinction Probabilities ◽  
Collision Processes ◽  
Necessary And Sufficient

We consider the uniqueness and extinction properties of the interacting branching collision process (IBCP), which consists of two strongly interacting components: an ordinary Markov branching process and a collision branching process. We establish that there is a unique IBCP, and derive necessary and sufficient conditions for it to be nonexplosive that are easily checked. Explicit expressions are obtained for the extinction probabilities for both regular and irregular cases. The associated expected hitting times are also considered. Examples are provided to illustrate our results.

Geometric rate of growth in population-size-dependent branching processes

1984 ◽  
Vol 21 (01) ◽  
pp. 40-49 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Population Size ◽  
Process Model ◽  
Branching Process ◽  
Necessary Conditions ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Size Dependent ◽  
Offspring Distribution ◽  
The Mean

We consider a branching-process model {Zn }, where the law of offspring distribution depends on the population size. We consider the case when the means mn (mn is the mean of offspring distribution when the population size is equal to n) tend to a limit m > 1 as n →∞. For a certain class of processes {Zn } necessary conditions for convergence in L 1 and L 2 and sufficient conditions for almost sure convergence and convergence in L 2 of Wn = Zn/mn are given.

Two necessary conditions for embeddability of a Galton–Watson branching process

1975 ◽  
Vol 78 (2) ◽  
pp. 339-343 ◽  
Author(s):  
D. R. Grey
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Substantial Contribution ◽  
Embedding Problem ◽  
Probability Generating Functions ◽  
The Subject ◽  
Complete Bibliography

1. Introduction and statement of results. The most substantial contribution in recent years to the literature on the Galton–Watson embedding problem has been that of Karlin and McGregor (4) and (5), to which the readeris referred for a more complete bibliography of the subject. Their results and methods very much involve the analytic properties of the associated probability generating functions (p.g.f.'s), and most of the results they describe as ‘negative’ in the sense that they are sufficient conditions for non-embeddability. The two results here are similarly ‘negative’ but obtained from more direct probabilistic considerations.

Extinction of population-size-dependent branching processes in random environments

1999 ◽  
Vol 36 (1) ◽  
pp. 146-154 ◽  
Author(s):  
Han-xing Wang
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Size Dependent ◽  
Branching Model

We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Zn}n≥0 is associated with the stationary environment ξ− = {ξn}n≥0, let B = {ω : Zn(ω) = for some n}, and q(ξ−) = P(B | ξ−, Z0 = 1). The result is that P(q(̅ξ) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(ξ−) = 1) = 1) and for non-certain extinction (i.e. P(q(ξ−) < 1) = 1) are obtained for the model.

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